On classical and hybrid shadows of quantum states

Classical shadows are a computationally efficient approach to storing quantum states on a classical computer for the purposes of estimating expectation values of local observ-ables, obtained by performing repeated random measurements. In this note we offer some comments on this approach. We note that the resources needed to form classi-cal shadows with bounded relative error depend strongly on the target state. We then comment on the advantages and limitations of using classical shadows to simulate many -body dynamics. In addition, we introduce the notion of a hybrid shadow, constructed from measurements on a part of the system instead of the entirety, which provides a framework to gain more insight into the nature of shadow states as one reduces the size of the subsystem measured, and a potential alternative to compressing quantum states.